Asymptotic integration of delay differential systems

JOURNAL

OF MATHEMATICAL

Asymptotic

ANALYSIS

ANII

Integration 0.

APPLICATIONS

138, 311-327 ( 1989)

of Delay Differential

ARINO*

AND

I.

Systems

GY~RI+

*Deparrement de MathPmatiyues. L’nu:er.~~rCde Pau. Avenue de I’limwrsrti, 64000 Puu. Frunw ‘The Medical lJnirersig of‘ Szegcd. Mungar\ Submuted by Kennerh L Cooke Recwed

February 20. 1987

INTRODUCTION

In this paper we consider the asymptotic integration of a delay differential equation. This means that we are dealing with non-autonomous equations which are asymptotically autonomous, and we look for formulae for the solutions at large values of the independent variable. As a well-known example, we can mention d.x ;i;=P(t).(.Y(I)-.I(z--)),

where p is in L* or liP TIp( ds d k < 1. Asymptotically, (1) is like .?= 0. and indeed it has been proved in [ 1.41 that the solutions of ( 1) are asymptotically constant. Another example is ds -z=

-ax(t-r(t)),

where r(t) -+ 0, t -+ cc. The asymptotic equation is dev/dt = -ax(t) and it has been proved by K. L. Cooke [S] that if r is in L’ then the solutions of (2) are asymptotically of the form exp( --at) . Const. The equation under consideration here is ~=ni(t)+

L(t, x,),

for t 3

t,,,

(3)

where s(t) E R”, x, denotes, as usual, the function defined on [ --, 0] by x,(s) = x(t + s), -T
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ARINO AND GYiiRl

(H,) For each t, L(t, .) is linear continuous from C( [ -t, 01, [Wn)into Iw”; t + L(t, .) is continuous and IIL(t, .)[I is in L2(t,, +co). The problem of asymptotic integration of such equations has been studied by J. R. Haddock and R. Sacker [9] for a more particular model equation ~=(n+A(t))x(t)+B(t)x(t-r),

(4)

with A and B in L’(t,, +K). Haddock and Sacker proposed a study of (4) when trying to extend previous results by Hartman [12], Hartman and Wintner [13], Atkinson [3], and Harris and Lutz [ 111, notably for ordinary differential equations of the form g=

(A +A(t))

.x(t),

where A is in L2.

(5)

In [9], a first result was established for the scalar case. In view of this result, J. Haddock and R. Sacker conjectured an asymptotic formula for the vectorial case. The conjecture states that there exists a matrix function F, F(t) + 0 as t + +ccl, and for each solution x, a constant vector c and a function h ,f( t) + 0 as t + +co such that .r;(t)=[Z~+F(t)].exp(S:A(~)~~).[c+f(t)],

(6)

where A(t)=A+diag{A(t)}+diag{B(t)}.e-‘”. We may observe that such problems were investigated earlier in a different perspective by J. Ryabov [16], R. Driver [6], and I. Gyiiri [7,8]. In this paper we mainly prove a result very close to the Haddock and Sacker conjecture for Eq. (3) under more general assumptions than in [9]. Our formula differs from (6) essentially in that F(t) is replaced by a functional G(t) defined on the space C( [ - 22, t], KY). Of course, we still have G(t) --) 0 as t + +co. Our result can be described as follows: for each solution x of (3), there exist a constant vector c, a function vi with values in [w”, VI(t) -+ 0 as t + +co, and a function q2, v2(t) E C( [ -2r, t], [W”),qz(t) --+0 as t + +oo such that

(7)

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313

While this formula is generally less agreeable than (6), we have been able to establish a better one in the case that we call quasi-triangular. In this situation, the formula (7) holds with G(t) = 0. This covers the scalar case and yields then the same result as in [9]. The method used in our paper is very close to the techniques employed in the study of partial stability. A few fundamental results along these lines are collected in Section 1. Section 2 deals with the quasi-triangular systems and Section 3 discusses the general situation. Our treatment relies on changes of variables which in particular allow us to give an inductive proof of the result with respect to the dimension FL

1. FUNDAMENTAL RESULTS PROPOSITION 1.

Consider the equation

in rchirh 1 B( t, c) = 0 (i.e., B I‘.Fa balanced term as defined in [ 141); ds. \ (B(t, cp)- B(t, $)I Qj’ i h(t,s) 2-g I .I (H3) : for ever)’ cp,* absolutely continuous, and

I

0 h(t-s,s)ds< lim sup J r I-/

(H 4

1,

ly’t, cP)-PC4 $11~P(t)~Icp-~Ic~,r~-r.ol. R”)> wzthp(t) andP(t, 0) in L’.

Then for each solution x of (8), lim, - ~ x(t) exists. Moreocer, ,for t, large enough. and each c in R”, there exists a solution of (8) defined on [t, - 5, + x ). ,cith c as a limit at infinity. Remark 1. The functional: cp -+ lim I-+*. u(t,, cp)(f) (with usual notations) is continuous for each t 0; and Proposition 1 states that it is surjective if to is large enough. In fact, as to + +szo, it tends to the Dirac function cp+ cp(0). This proposition was proved in [ 11. The convergence was also established in [4]. In this last paper, a significantly more general condition

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ARlNOANDGYijRI

than the one in Proposition 1 was obtained by coupling in some way p(t) and h( t, s) (see also [ 11, for a discussion on such conditions). PROPOSITION 2.

Consider the equation

(9)

$ = P(t,x,) + Q(t,y,), N(resp. P): [to, + co) x C( [ -7, 01, R”) + R”(resp. Rm), N(resp. Q):

[to, + co) x C([

-5, 01, lRm) --+ R”(resp. KY),

where M, N, P, Q are continuous linearfunctionals variable. Assume moreover that (H,)

with respect to the second

the equation ;

= M(t, x,)

(10)

is stable; (Hb) the equation

~=g(r,Y,) is exponentially (H,)

(11)

stable;

IIN(t, .)/I and IIP(t, .)I/ are in L’.

Let (x, y) be a solution of (9). Then x is bounded and lim, _ ocy(t) = 0. Moreover, if for all the solutions u(t) of (lo), lim, _ ~ u(t) exists, the same holds with the solutions of (9). Remark 2. (a) We will not need Proposition this result is of independent interest.

2 in its full generality but

(b) We need to assume linearity of M and Q because we will use variations of constant formulas from Eqs. (10) and (11). But, a Lipschitzean nonlinearity in N and P is admissible. Proof of Proposition

2

Notation 1. We will denote by U(t, s) (resp. V(t, s)), t > s, the solution operator associated with (10) (resp. (11)).

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315

So U(t, s) . s, = X, for each solution .Yof ( IO), and I’( t. s). J‘\ = .I’, for each solution 1’ of (11). From (H,) and (H6) (in Proposition 2), we deduce I U(t, .s)l Q K,

(V(t,s)l~K.exp[-~(t-.s),

for some cc>O, K-c +x

t >, s,

[lo].

(121

We will express the assumption (H, ) in the form

IN(t, cp)l
,.:

IOr. $11
with n and p in L’.

(131

Let (s, y) be a solution of (9). Using the variation of constants formula for (10) and (11) we can represent x,, .v, as

.y,= ut, to) s,,,+ j ’ rq t. s ) .Y,,N(.s,J‘,) ds, ,,I

(14)

.I’, = V(r. 1,) j‘,, + ’ I’( t, s) Y,, P( s, s, ) ds, sl,,

where X0 (resp. Y,) denotes a function with a bounded variation associated to the Dirac distribution at 0 in R” (resp. RR’). Define Variuhle 1. u(t)= 1.~~1, t(t)=

Ij’,(.

From ( 12) and ( 14) we can see immediately that (u. I‘ ) verities a system of inequalities: MGKu(z,)+KJ’~(~)~(~)~.~, I,,

(15)

f tl( t) < Ke z”-‘“‘t~(tO)+K

r e “‘~ “p(.~)~(.~)d~, i I,,

We will show first that u is bounded and r(t) -+ 0 as t -+ z. Combining the two inequalities of (15) we can derive an inequality involving only 11(resp. only 11).This gives u(t)
/‘n(s)e-~‘I’ to ‘e

‘I’

‘“Ids “p(r)u(S)dT

(161

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ARINO AND GYiiRI

Notation 2.

Let

h(t,, uo, oo)=Ku,+K2 c(t,)=K2.~‘x

jx n(s)e-““-‘“‘ds ( 10

m

n(s)

I‘ePxl’ f0

LEMMA 1. h and c are uniformly c(to) -+ 0 as t, -b +~0.

1

.uO,

for uO,u03 0,

-“p(z)dz bounded bcith respect to t,,. Moreover

These two assertions are easy consequencesof the Schwartz inequality in L2 and convolution product properties of L2 by L’ functions.

We can now continue the proof of the proposition. In view of Notation 2 the above inequality (16) can be written as

u(t) d Mt,, 4t,), u(t,)) + c(t,). rg<,
(17)

which implies the same with u(t) changed into max,,. ,<, u(s) on the left hand side. So, if t, is large enough, c(t,) will be < 1 (from Lemma l), and then we get

Gto)) u(t) s Mb 14klh -c(to) ’

t3 1,.

(18)

Using the same arguments with + cc replaced with t, + T for T small enough (independently of to), we can prove that max 24(s)6 C . max u(s), $
We come now to the last part of Proposition 2. In this part, we assume that U( t, s) cpconverges as t -+ +c;o. Notation

3.

U,(s)=lim,,

+,rJ U(t, s).cp.

We only have to look at formula (14). The first term tends to .x,,: the term under the integral converges pointwise to

U,(t,)

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DELAY DIFFERENTIALSYSTEMS

U,(s) X0. N(s, y,). But since N(s, y,) is in L’ (as we noticed in Remark 3) and U(t, s) X,, is bounded, the convergence is dominated. So the right side in (14), has a limit at infinity, which means that x(t) converges and we have ,‘ijy x(t)=

U,(t,,)x,,+~

_- U,(s) X,,N(s,y,) 10

ds.

(19)

We will consider now a more specific version of Proposition 2. Assume (He), (H,) and that IM(t, .)I is in L’. Let (x,J) be a solution of (9). Then lim,, x x(t) exists, lim,, I y(t)=O. Moreover, if t, is large enough, for each c in NV’ there exists a solution dqfined on [t,,, + co) such that lim,, +,Xx(t) = c. COROLLARY 1.

Proof: From Proposition 1, we deduce immediately that Eq. (10) is stable and its solutions converge. So, all the conditions of Proposition 2 are satisfied, which yields the first part of Corollary 1. Moreover, lim, _ x .u(t ) is given by the right side of (19). We noticed in Remark 1 that U,(t,) . cp-+ q(O), as t, + +xl. At the same time, the integral tends to 0 (in the sense of norm on such functionals). More precisely, we have, using ( 15) variable 1,

So, in view of Notation 2, we obtain an estimate of the integral of the form ~(2~). (Ix,,, / + 1y,,,I), where .s(tO)+ 0, as t, -+ + co. Therefore, if we restrict our attention to constant data x,, = x0 and yro= 0, we will obtain lim x(f,, .x0, O)(t) =.Y” + f--t +x

0(x0)

for t, large enough. This gives the desired surjectivity.

2. CASE OF A QUASI-TRIANGULAR

MAP

We consider now Eq. (3). In addition to (H, ) we assume from now on that the 2,‘s are ordered: for i < j, 2, > A,. This assumption does not restrict the generality. We will frequently use a matrix representation for L, namely L(h .)= W,(c .))I<,.,<“’

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ARINOANDGYijRI

and, for a matrix A, we will denote by diag{A} constituted with diagonal elements of A. THEOREM

the diagonal matrix

1. Assume (Hi), (Hz), and (H,) hold, where someE> 0 and each i, j, i > j, there exists C > 0, .)I1~C.exp((l,-3L,-&)t))).

Thenfor every x solution of (3) there exists a constant c in KY’such that x(t)=exp(rA(s)ds).[c+o(l)], where A(t) = A + diag(l(t, exp(A.))}. The functional q --+lim, _ m exp( -f’ A(s) ds) .x(t,, q)(t) is continuous, and surjective if I,, is large enough. Remark 4. (i) Obviously, this theorem applies to the triangular case, and, in particular, to the scalar case. (ii) The result obtained here is stronger than the one conjectured in [9] since we have F(t) = 0 (see Introduction). The proof is based on two changes of variables which will reduce Eq. (3) to an equation of type (8) and on the application of Proposition 1 to the transformed equation. First Change of Variables Variable 2. y(t) = exp( -At) .x(t). The first expression for the equation on y is simply $=L(t,

Y,)==~exp( -At) .L(t, exp A(t + .) . y,)

(21)

where

E&t, .I = exp - &t-L&t, exp(A,(t+ .)I. ( 1,) =exp(l,-U (We represent the map s+exp(as) estimate of 2, in terms of L,:

t.L,(t, (eq$.).(

),).

by exp(a.)). This leads to a first

II~,(t, -)I1G C,eXp(Aj-1,) t. IIL,(t. .)II with C,=max-,.,.,exp;l,s.

(22)

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Notation 4. q = max({A, - A,; i
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SYSTEMS

-E)

(E as in

(HE)),

I(t) =

In view of the ordering on the 2,‘s and (Hz), we have q < 0 and 1 in L’, and in view of (22) and (H8), we obtain new estimates for l,,, namely

II&(4 .)I1=0(@‘.4t)), IILk

for i#j

and (23)

.)II = W4t)).

These estimates suggest another presentation of Eq. (21) that is (24)

where d(t, .)=diag{Z(t, .)} is . such that 11 A( t, ) II is in L’, and, therefore, q’. E(t)) is in L’. We rewrite A in the form IIN6 )I1= We

A(t, cp)= A(& do)) + A(4 cp- cp(O)). We have A(t, cp(O))=diag{~?(t, cp(0))). So, in view of Eq. (21) and the definition of A(t) given in the statement of Theorem 1, A(t, q(O) = diag{l(t,

exp A.)} . q(O)

=(n(t)-n)~rp(O). Notation

5. ;?(t) = A(t) - A.

Notation

6. A,(& cp) = A(t, cp- q(O)).

A, is still diagonal, but, because of the presence of cp- q(O), it can be expressed as a functional on C( [ -2r,O], R’), for t B to + z, using y, -y(t) = s; + dy/ds . ds, and replacing dy/ds with the right side of (24). We get

IA,(t,y,)l Q IlA(t, .)II . j’

II&, .)II ds+s’ Ilk 1--T r-x

.)ll /Js (25)

and so

IIAI(c .Nc+2r,01. Rn)is in L’.

(26)

For the rest of the proof, we will consider that A i is acting on C( [ - 2t, 01) and the notation y, will correspond to the translation over [ -22,0].

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ARINO

AND

GYiiRI

In terms of A r, the equation reads as

$= A,(&Y,)+ &I .y(t) + h(t,Y,).

(27)

Second Change of Variables Variable 3. z(t)=exp( -s’J?(s) ds) .y(t). Using (27), we obtain the following equation in z: $=exp(-~~(s)ds).A,(r,exp(~+

/7(s)ds).z,)

+exp(-rA(,)ds).h(r,exp(r+‘n(,)ds).z,). Because A, is diagonal, the first term in the right side of (28) reduces to A2(t, z,), where Notation

7. A,(& z,) = A,(& exp(~:+‘~(.s) ds) .z,).

From (26) we see that IIAz(t, .)[I is in L’.

(29)

Notation 8. h2(t, cp) = exp( -jr A(s) ds) . h(t, exp(j’+. A(s) ds) . cp). Because A(s) is in L*, we have I/j’ A(s) dsll = O(fi), and therefore

lh2(t, cp)l G exp(Cl .,h). C2 .e’l’ .1(t) exp(C, fi). l(t). O(e (n/2)f), which implies in particular that Ilh2(t, .)I[ is in L’.

Id.

So

IlhAt, .)II = (30)

In terms of A,, h,, Eq. (28) reads as ;

=

A2(4

z,) + hz(t, z,),

(31)

which, in view of (29) and (30), verifies the assumption (H4) of Proposition 1. So, from Proposition 1, it follows first that z(t) = c + o( 1), and, coming back to X, we get the formula (20) of Theorem 1. The continuity of the limiting functional is also a consequence of Proposition 1 and Remark 1, as well as the surjectivity. But, surjectivity holds with respect to the set of data for Eq. (31), which is C( [ -2r,O], KY). The solutions of (3) constitute only a subset of this set. We must then show that there is still surjectivity with respect to the solutions of (3) or, equivalently, to the solutions of (21). The reason for this is to consider

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321

SYSTEMS

special data for (21) (as we did when considering the surjectivity for Corollary 1). We take yfO= exp(j”+ A(s) dr) .c, which, by the transformation induced in Variable 3, leads to a set of data for (31) of the form

Z/[&-T, t,]=c

and

z/[t,,

to + tl = exp

.y/[t,,

f. + t].

Denote by c”= z/[to - z, t, + r]. If t, is large enough we can prove that F will be close to c. On the other hand, we know (see Remark 1) that, if we denote by z(t,, .)(t), this functional tends (uniformly with =(t,, .)(co)=lim,,, respect to the bounded sets) to 6,, so that (1) z(t,, . )( co) is uniformly bounded (in to) and (2) z(t,, c)(co) = c + ~(t,,) . c, with lle(to)il + 0, as t, -+ +CQ. Using (1) and (2) we see that for t, large enough lz(t0+T, F)(m)1 2 (cl - le(z,+z)l .Icl -Iv.

(c--T1 >o,

for (cl = 1.

This implies the surjectivity with respect to the restricted space,and, so, the surjectivity of the limiting functional associated with (3) which completes the proof of Theorem 1. 3. THE GENERAL CASE

We now assume only (H,) and (Hz). Using Proposition 2 and its corollary, we will prove the following: THEOREM 2. Assume (H,) and (Hz). Then, there exists a family of matrix-ualuedfunctionals ~(t, t,, .)= (.e,/(t, to, .))lGr,,Sn, t > t, such that

(i) (ii) (iii) (iv)

eV=O,for jai; ~,(t, t,, .) is a bounded functional

on C( [t, - 22, t], UP);

(I&&f, to, =)I1+O as t+ +co (or, as t,-+ +oo); t + I(s,(t, t,, .)I1 is in L*(t,, + ~0);

and for every solution x of (3) on [to, + 00) there exists a constant c in KY’ such that

>I.(c+o(l)),

’ A(s) ds

t >, to+ 22, (32)

where 6, denotes the Dirac distribution (or, the evaluation map) at t, that is 6,((p) = q(t); A(t) is as in Theorem 1. The functional x,, -+ c is continuous and surjective tf t, is large enough. 409!138’2-3

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ARINOANDGYijRI

Remark 5. While the formula (32) follows naturally from the proof of the theorem, it may appear strange at first. We can express it in a more explicit form which refers us to formula (7). For that, consider for simplicity that t, = 0. Denote by G(t) = s(t, 0, .) . G(t) is a functional on C( [ -22, r], Rn) such that IIG(t)ll --, 0 as t + + co. Now, the theorem says that for each x solution of (3), there exists a constant (’ and a function ~(t, .)EC([-22r, t], R”), Ilq(t, .)I/ +O as t+ +co, such that

‘.

(33)

Prooj The first step of the proof is to write (3) under the form of (9). We decompose the variable x = (x, , z), X, E IQ,z E IX”+ ’ and change the variables. Vuriuble 4. u(t)=exp(-J’I,(s)ds).x,(t); y(t)=exp(-S’I,(s)ds).z(t) (where 1,(t) = n,,(t)). (u, y) satisfies the equation

2 = 46 u,)+ PC6 Y,), (34)

~=Y(w,)+w,Y,)LEMMA 2. cr(t, .) (resp. P(t, .)) are bounded linear jiinctionals on C([-2~,0], [w) (resp. C([ -2r,O], W-l)). Moreouer I(cr(t, .)/I is in L’ and II/3(t, .)I/ is in L2.

Let us prove these results. After changing the variables (x, z) into (u, y), the equation in u takes the form

(35)

The first group of terms in (35) corresponds to cr(t, u,) in (34) while the other which depends only on y corresponds to j?(t, yl). For this last term, we can see that it satisfies the properties stated in Lemma 2, that is, I[/?(& .)I[ is in L2.

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323

We only have to consider the terms in u. We observe that

and t-‘S;+\L,,(H, ei., ) d6, is in L’ and L’ , uniformly with respect to s E C-r, 01. We will denote this fact by writing this function as O(L’nL’). So expj:’ E.,(s)ds=exp(E,,t).(l +O(L”nLL’)). Then the first term in (35) can be written as L,,(t,exp(i.,.).u,)+a,(f,

II,),

where a,(& .) verities the desired property that l(~,(t, .)\I is in L’ (and. in fact, lla,(t, ,)I\ is also in L’). So, we have to concentrate on L,,(t,exp(A,.)(u,-u(t))), and we will express II, - u(t) in terms of the derivative of u, and, then of u/[t - 2s, t]. Using (35). we have I(, - I((t) = j; + du/ds ds, so that L,,(r,exp(E+,.).(u,-u(t)) = L,,(t,exp(i,.).{‘+

rI+

I

L,l(~,ex~(~w,-)~(u,-u(s)))ds+L,,(t,exp(i,.)

L,,(~,0(L’nL”).u,)ds+

i’

L,,JI’+

L,,,(s,ev

f (1:’

L,,(t,exp(i.,.)

;).11.,).)ds].

Once again using arguments on products of Lp, Ly functions and the fact that the operator j:+ sends Lp into Lp n L L for each p 2 1, we seethat the first operator in the right side has its time-dependent norm in L’, as well as the second one, and the third one is in L’ n L2. Thus, the proof of Lemma 2 is complete. We turn now to the consideration of y( t, u,) and 6( f, I’,) in (34). LEMMA

3.

Ily(t, .)I\ is in L2, and the equation $ = S( t, vr) is exponentially

stable.

(36)

The proof of this lemma is straightforward. We only have to express y and 6 in terms of the original equation. Immediately from the change of variables, we get

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so that Ilv(r, .)I[ is in L’ and d(t, cp)= (A’i,(t) I’) q(O) + L’ t, exp j’+ I,(s) ds .cp ( ( , > > (where we denote by the prime the restriction of operators to Rn-‘). A’ - i,(t) I’ is diagonal with ith elements A,-/I-L,,(t,eL’),

i = 2, .... n,

1, - 2, < 0, and L, ,(t, e”’ ) is an L*-perturbation, as well as the last term in the right side of d(t, cp).This ensures the exponential stability of (36). The conclusion from these two lemmas is that (34) verifies the conditions of Proposition 2, and, more specifically, its corollary. Therefore the conclusions of Corollary 1 hold: lim y(t)=0 I--r +r

and

lim u(t) exists. , - +‘cc

This limit is a continuous functional of the data of the original equation (3). Concerning the surjectivity, the same problem as in Theorem 1 arises. Surjectivity holds, for to large enough, with respect to the data in C( [ -2r,O], Rn). A similar argument proves that it holds also with respect to C( [ - T, 01, W). We come back now to the original variables (x,, z). We will denote by V(t, s) (as in Notation 1) the solution operator for Eq. (36). We get a first expression for (x, , z), X,(t)=exp(lii,(r)ds).[c,+o(l)], ;(l)=exp(J’i,(r)ds).[

(37)

1

v(t, to) .y,,(O) + j’ ( J’(L s) Y,)(O) .Y(s, u,) ds f0 (38)

(where c, = lim,, +nou(t)). The right side of (38) can be decomposed into a sum: z = z”‘(t) + z”‘(t). Variable 5. z”‘(t) = exp(J’ n,(s) ds) s:, (V(t, 3). Y,)(O) y(s, u,) ds. Notation 9. For t> to, we denote by zl(t, to, .) the following functional from C([t,-22, t], R) into R”: sI =(~r,) where .sll =0 and

S)*Yo)j(O) Y(S, Cps) dsvj = 2, .... n. El,(t, to,CP) =s10’ (V(t,

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LEMMA 4. e,,,j= 1, .... n, verifies Theorem 2.

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the propertres

(i t(iv)

stated

in

Proof. (i) follows from Notation 9; (ii), (iv) are direct consequencesof the estimate

which is due to (12), and the fact that /ly(s. )I1 is in L’ (Lemma 3). In terms of a,, ,-‘?’ is given by ;‘~‘(r)=&,(r,f~,-)(expS’I,(s)ds).[c,+o(l)] /

and (VA to) ~.v,,,)(O)).

(41 I

This means that 2“’ is a solution of

$=nk(t)+L’(l..-,),

(42)

where, as in Lemma 3, A’, L’ correspond to the restriction of A, L to I&!” ’ We are now ready to complete the proof of Theorem 2. After the first step, the problem has been reduced to a problem of the same type with one dimension less. This suggestsan iterative procedure and the formulas (37) and (40) give an indication of what will be the general expression. In fact, assuming that we can find an expression like (32) for (42) we will then have ~“‘(f)=(fi,+&‘(t,

1”, -))exp(i;l’(s)ds).[r’+o(f)].

t>t,+2t.

(43)

Combining (37), (40), and (43) we obtain (32) for X= (s,. z). The proof of Theorem 2 is complete. In conclusion, we emphasize a very striking difference between the results obtained here (which are more or less those conjectured in [9]) and the asymptotic integration results in the ordinary case. In [ 131, Hartman and Wintner recalled a first theorem in this field due to 0. Perron [15]: assuming that A(t) (in Eq. (5)) tends to 0 at infinity and A has n distinct eigenvalues, (i,)r=l, ,n; (5) has n independent solutions (x,). with

Ilx,(r)ll - e’l’,

t+ +r_.

326

ARINO AND GYiiRI

They obtained extensions of this result, still with the existence of n different exponential growths. In [l l] Harris and Lutz consider also an ordinary equation and show, under certain conditions, the existence of a fundamental solution of the form Y(r)= [Z+0(1)]expS’n(s)&

(44)

This formula also implies the existence of n distinct exponential growths. In contrast to that, even in the triangular case, the results we get do not allow us to separate different asymptotic exponential growths. The best interpretation we can make in this direction is to say that each solution can be written as a combination of functions with distinct growths. The delay seems to prevent formulas such as (44). There are some cases however where we can at least find solutions with distinct exponential growths, in particular, the case where the right side of (3) is “small enough.” We will develop this assertion now. We know that under an appropriate smallness condition (3) has an n dimensional subspace of “special” solutions, defined on 88,uniquely determined by their value at a point [16, 7, 8, 1, 23. We established in [2] that these solutions are associated to an ordinary differential equation. The “smallness condition” is that, if we denote M(t, cp)= n&O) + L(t, cp), we have IIM(t, .)[I
REFERENCES 1. 0. ARINO AND I. GY~RI, Asymptotic integration of functional differential systems which are asymptotically autonomous, in “Pubhcations Mathematiques de I’Universiti de PAU” and in “Proceedings of EQUADIFF. 82,” pp. 37-59, Lect. Notes, Vol. 1017, SprmgerVerlag, Berlin/New York, 1986. 2. 0. ARINO AND I. GY~RI, Differential systems asymptotically equivalent to ordinary differential systems, in “Qualitative Properties of Differential Equations” W. Allegreto and G. J. Butler Eds.), pp. 27-37, Proceedings of the Edmonton Conference, 1987.

DELAY DIFFERENTIAL 3. F V ATKINSON, The Asymptotic Puru Appl. 31 (1954), 347-378. 4

5. 6 7. 8

9.

10. 1I

12. 13 !4 15. 16

SYSTEMS

solution of 2nd order dlfferentlal

327 equations, Ann. Mur

F V ATKINSON AND J. R HADDOCK, Criteria for asymptotic constancy of solutions of functlonal differential equations, J Mafh. Anal. Appl. 91 (1983). 41CU23. K L. COOKE. Asymptotic theory for the delay-chfferential equation u’(r) = au(r - r(u( t) I ). J Ma/h. Anal. Appl. 19 (1967). 160-173. R D DRIVER, Lmear &tTerentlal systems with small delays. J Dlf/erentral Equatumc 21 (1976). 147-167 I GY~RI. On asymptotically ordinary functional differential equations, UI “Colloq. Math. Sot. Janos Bolyal 15, DdT Eq.,” pp 257-268. North-Holland. Amsterdam 1977. I GvORl, On existence of the limits of solutions of functional differential equations. In “Colloq. Math. Sot. Janos Bolyal 30, Qual. Theory of Diff. Eq.,” pp. 325-367. North-Holland, Amsterdam, 1980. J R HAUIIO~K AND R. SACKER, Stability and asymptotic mtegratlon for certam lmear systems of functional dlfferentlal equations, J. Math. Anal. 76 ( 1980). 328-338. J K HALE. Theory of functIonal chfferentlal equations. ~1 “Applied Mathematical Sciences 3.” Sprmger-Verlag, New York,iBerhn, 1977. W A. HARMS AND D. A LUTZ, A unified theory of asymptotic mtegratron. J Murh 4d A~jp/ (1977). 571-586 1964 P HAKTMAN, ‘*Ordinary Dlfferentlal Equations,” Wiley, New York/London/Sydney. P HAKTMAN AND A. WINTNER. Asymptotic integration of hnear chfferentlal equations. .4nwr J. Math. 77 (1955 ), 45-86. P. S PANKOV. Asymptotic finite dlmenslonahty of the space of solutions of a certam class tif systems with lag. D$ferenfslal ‘n)v Uraonenqu 13 (1977). 455462 [Russian] 0 ERRON, tiber Stabilitad und asymptotisches Verhalten der Integrale von DIfferentlaiglelchung systemen, Math. Z. 29 (1969). 129-160. JC A RYABOV, Certam asymptotic propertIes of linear systems with small time lag. jr;udl, &WI Tear DIG’ Dru:hx Narodoc Patri.ya Lumumhr 3 (1965 ). 153-165 [Russian]